We often have such confusion: not only have we talked about it many times, but students' problem-solving ability just can't be improved! I often hear students complain that the consolidation problem has been done thousands of times, but the math scores have not improved! This should arouse our reflection. It is true that the above situation involves all aspects, but the example teaching is worth reflecting. Mathematical examples are a key step from the generation to the application of knowledge, which is also called "throwing bricks to attract jade". However, in many cases, examples follow examples and do not guide students to reflect after solving problems, so students' learning still stays at the surface of examples. This is not surprising.

Kong Ziyun: Learning without thinking is useless. "Nothing" means confusion and no gain. By extending its meaning, it is not difficult for us to understand why the example teaching should reflect after solving the problem. In fact, post-solution reflection is a process of summing up knowledge and refining methods; It is a process of learning lessons and gradually improving; This is a process of harvesting hope. From this point of view, the post-solution reflection of example teaching should become an important content of example teaching. This paper intends to discuss from the following three aspects.

First, reflect on the law of solving problems.

It is difficult to achieve the purpose of improving problem-solving ability and developing thinking. Being good at reflecting after solving problems, classifying methods, summing up laws and trying to figure out skills, so as to make a question changeable, ask more questions and solve more questions, tap the depth and breadth of examples and expand the radiation surface of examples, which is undoubtedly conducive to the improvement of ability and the development of thinking.

For example, it is known that the waist length of an isosceles triangle is 4 and the base length is 6; Find the circumference. We can change this example.

Variant 1 It is known that the waist length of an isosceles triangle is 4 and the circumference is 14. Find the base length. (This is to test the ability of reverse thinking)

Variant 2 has an isosceles triangle with a side length of 4; The other side is 6. Find the circumference. (Compared with the first two questions, we need to change our thinking strategies and conduct classified discussions. )

Variant 3: It is known that one side of an isosceles triangle is 3 and the other side is 6. Find the circumference. (Obviously, "3 can only be the bottom" is inconsistent with the fact that the sum of two sides of a triangle is greater than the third side, which is conducive to cultivating students' rigorous thinking)

Variant 4: Given the waist length of an isosceles triangle as X, find the range of the base length Y. ..

Variant 5: It is known that the waist length of an isosceles triangle is X, the base length is Y, and the circumference is 14. Please write the functional relationship between them first, and then draw their images in plane rectangular coordinates. (Compared with before, the requirements have been improved, especially the understanding and application of the condition 0 < y < 2x is the key to complete this question. )

Another example is: Example 2 on page 93 of the third grade geometry of People's Education Press is different from the example on page 107 1, which is a rare material for solving problems (AB is the diameter ⊙O, C is a point on ⊙O, AD is perpendicular to the tangent passing through point C, and the vertical foot is D. Verification: AC split ∠.

Through the layer-by-layer variation of examples, students' understanding of the theorem of trilateral relations has deepened, which is conducive to cultivating students to analyze and solve problems from special to general and from concrete to abstract; Problem-solving variant teaching is helpful to help students form and break the fixed thinking mode; It is beneficial to cultivate the flexibility and flexibility of thinking.

Second, students are prone to make mistakes in reflection.

Students' knowledge background, way of thinking and emotional experience are often different from those of adults, and their expressions may be inaccurate, which will inevitably lead to "mistakes". If example teaching can start from now and reflect after solving it, we can often find the "root cause" and then prescribe the right medicine, which can often get twice the result with half the effort!

There is such a case, published in the fifth issue of Mathematics for Primary and Secondary Schools in the junior middle school (teacher's edition) in early 2004: The question asked by a senior one teacher: -3× (-4) =? A student's answer is "9", and the teacher looks at it: wrong! So I immediately asked student B to answer, and the student's answer was "12", so the teacher asked him to explain the algorithm: …… After class, the teacher interviewed the student who gave the wrong answer, and the student said: Stand at the point of -3, because you multiply by -4, so you have to move four times in the opposite direction along the axis of the number, three squares at a time, so the answer is 9. His answer is really wrong. How come? Why do you have such an idea? How to correct it? If our example teaching can seize this opportunity to discuss and reflect, it is undoubtedly much better than telling ten, a hundred or even more examples to consolidate the law, which we easily ignore.

Calculation is the key and difficult point of algebra teaching in senior one. How to grasp this key point and break through this difficulty? Teachers can be described as "doing everything they can" in example teaching. For example, after completing the nature of power and entering the next stage-multiplication and division of monomial and polynomial, the author designed the following two examples:

(1) Please point out the meanings of (-2) 2, -22, -2-2 and 2-2 respectively;

(2) Please distinguish the following categories:

① a2+a2=a4 ②a4÷a2=a4÷2=a2

③-a3 (-a)2 =(-a)3+2 =-a5

④(-a)0÷a3 = 0⑤(a-2)3a = a-2+3+ 1 = a2

After the solution, the author will guide the students to reflect and summarize.

What are the common mistakes in the calculation of (1)? (2) What are the reasons for these errors? (3) How to overcome these mistakes? Students express their opinions and prescribe effective prescriptions for various "causes". Practice has proved that this example teaching is successful, and students greatly improve the accuracy and speed of calculation.

Third, reflect on emotional experience.

Because the whole problem-solving process is not only a process of knowledge application and skill training, but also a comprehensive process accompanied by communication, creation, pursuit, joys and sorrows, and the participation of students' whole inner world. In the meantime, he not only tasted the bitterness of failure, but also gained the joy of "suspecting no way back, and having a bright future". He may have thought independently or solved it through cooperation and coordination, which not only embodies the value of personal efforts, but also reflects the light of collective wisdom. Guiding students to reflect after solving problems here is conducive to cultivating students' positive emotional experience and learning motivation; It is beneficial to stimulate students' interest in learning, ignite their enthusiasm for learning, and change passive learning into independent inquiry learning; It is also conducive to cultivating students' learning perseverance and will. At the same time, in this process, students' study habits of independent thinking, sense of cooperation and team spirit can be well cultivated.

Friedenthal, a mathematical educator, pointed out that reflection is the core and motive force of mathematical activities. In short, the methods and laws of post-settlement reflection have been summed up in time; After the solution, the reflection made us get rid of the mystery, see the true face of Lushan Mountain, and gradually mature. In reflection, I learned to think independently, to listen, to communicate, to cooperate, to share, and to experience the fun of learning and the comfort of communication.

For more math papers, please see the math papers column.

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